Entropy solution

Remarks

Take $k$ larger than $\sup \abs{u}$ and $\sup \abs{g}$, and then smaller than $-\sup \abs{u}$ and $-\sup \abs{g}$. This gives two opposite inequalities whose combination says that $u$ is a weak solution of the initial value problem. So every entropy solution is a weak solution. Kruzhkov's theorem states that for each bounded measurable $g$ there is exactly one entropy solution, and that it depends continuously on $g$ in $L^1_{\mathrm{loc}}$. Let $u$ be piecewise smooth in one space dimension, with $f$ strictly convex. Then the condition is equivalent to the Rankine–Hugoniot jump condition together with the Lax condition $u_- > u_+$ across each shock. Here $u_-$ and $u_+$ are the values to the left and to the right of the shock.

Example: for Burgers' equation, $n = 1$ and $f(u) = u^2/2$, with $g = 1$ for $x < 0$ and $g = 0$ for $x > 0$, the shock $u = 1$ for $x < t/2$ and $u = 0$ for $x > t/2$ is the entropy solution. Its speed $1/2$ is $(f(1) - f(0))/(1 - 0)$, as Rankine–Hugoniot requires, and $u_- = 1 > 0 = u_+$.

Non-example: with $g = 0$ for $x < 0$ and $g = 1$ for $x > 0$, the shock $u = 0$ for $x < t/2$ and $u = 1$ for $x > t/2$ is a weak solution but not an entropy solution, because $u_- = 0 < 1 = u_+$. The entropy solution is the rarefaction fan $u = 0$ for $x < 0$, $u = x/t$ for $0 \le x \le t$, and $u = 1$ for $x > t$.